Scilab Function
Last update : 11/3/2005

splin2d - bicubic spline gridded 2d interpolation

### Calling Sequence

  `C = splin2d(x, y, z, [,spline_type])`

### Parameters

- `x,y`: strictly increasing row vectors (with at least 2 components) defining the interpolation grid
- `z`: nx x ny matrix (nx being the length of `x` and ny the length of `y`)
- `spline_type` : (optional) a string selecting the kind of bicubic spline to compute
- `C`: a big vector with the coefficients of the bicubic patches (see details in Remarks)

### Description

This function computes a bicubic spline or sub-spline s which interpolates the (xi,yj,zij) points, ie, we have s(xi,yj)=zij for all i=1,..,nx and j=1,..,ny. The resulting spline s is defined by the triplet `(x,y,C)` where `C` is the vector (of length 16(nx-1)(ny-1)) with the coefficients of each of the (nx-1)(ny-1) bicubic patches : on [x(i) x(i+1)]x[y(j) y(j+1)], s is defined by :

                 _4_   _4_
                 \     \                   k-1       l-1
        s(x,y) = /     /    C (k,l) (x - xi)  (y - yj)
                 ---   ---   ij
                 k=1   l=1

The evaluation of s at some points must be done by the `interp2d` function. Several kind of splines may be computed by selecting the appropriate `spline_type` parameter. The method used to compute the bicubic spline (or sub-spline) is the old fashionned one 's, i.e. to compute on each grid point (xi,yj) an approximation of the first derivatives ds/dx(xi,yj) and ds/dy(xi,yj) and of the cross derivative d2s/dxdy(xi,yj). Those derivatives are computed by the mean of 1d spline schemes leading to a C2 function (s is twice continuously differentiable) or by the mean of a local approximation scheme leading to a C1 function only. This scheme is selected with the `spline_type` parameter (see `splin` for details) :

  - "not_a_knot": this is the default case.
  - "natural"
  - "periodic": to use if the underlying function is periodic : you must have z(1,j) = z(nx,j) for all j in [1,ny] and z(i,1) = z(i,ny) for i in [1,nx] but this is not verified by the interface.
  - "monotone"
  - "fast"
  - "fast_periodic"

### Remarks

From an accuracy point of view use essentially the not_a_knot type or periodic type if the underlying interpolated function is periodic.

The natural, monotone, fast (or fast_periodic) type may be useful in some cases, for instance to limit oscillations (monotone being the most powerfull for that).

To get the coefficients of the bi-cubic patches in a more friendly way you can use `c = hypermat([4,4,nx-1,ny-1],C)` then the coefficient (k,l) of the patch (i,j) (see equation here before) is stored at `c(k,l,i,j)`. Nevertheless the `interp2d` function wait for the big vector `C` and not for the hypermatrix `c` (note that one can easily retrieve `C` from `c` with `C=c(:)`).

### Examples

    // example 1 : interpolation of cos(x)cos(y)
    n = 7;  // a regular grid with n x n interpolation points
            // will be used
    x = linspace(0,2*%pi,n); y = x;
    z = cos(x')*cos(y);
    C = splin2d(x, y, z, "periodic");
    m = 50; // discretisation parameter of the evaluation grid
    xx = linspace(0,2*%pi,m); yy = xx;
    [XX,YY] = ndgrid(xx,yy);
    zz = interp2d(XX,YY, x, y, C);
    emax = max(abs(zz - cos(xx')*cos(yy)));
    xbasc()
    plot3d(xx, yy, zz, flag=[2 4 4])
    [X,Y] = ndgrid(x,y);
    param3d1(X,Y,list(z,-9*ones(1,n)), flag=[0 0])
    str = msprintf(" with %d x %d interpolation points. ermax = %g",n,n,emax)
    xtitle("spline interpolation of cos(x)cos(y)"+str)

    // example 2 : different interpolation functions on random datas
    n = 6;
    x = linspace(0,1,n); y = x;
    z = rand(n,n);
    np = 50;
    xp = linspace(0,1,np); yp = xp;
    [XP, YP] = ndgrid(xp,yp);
    ZP1 = interp2d(XP, YP, x, y, splin2d(x, y, z, "not_a_knot"));
    ZP2 = linear_interpn(XP, YP, x, y, z);
    ZP3 = interp2d(XP, YP, x, y, splin2d(x, y, z, "natural"));
    ZP4 = interp2d(XP, YP, x, y, splin2d(x, y, z, "monotone"));
    xset("colormap", jetcolormap(64))
    xbasc()
    subplot(2,2,1)
       plot3d1(xp, yp, ZP1, flag=[2 2 4])
       xtitle("not_a_knot")
    subplot(2,2,2)
       plot3d1(xp, yp, ZP2, flag=[2 2 4])
       xtitle("bilinear interpolation")
    subplot(2,2,3)
       plot3d1(xp, yp, ZP3, flag=[2 2 4])
       xtitle("natural")
    subplot(2,2,4)
       plot3d1(xp, yp, ZP4, flag=[2 2 4])
       xtitle("monotone")
    xselect()

    // example 3 : not_a_knot spline and monotone sub-spline
    //             on a step function
    a = 0; b = 1; c = 0.25; d = 0.75;
    // create interpolation grid
    n = 11;
    x = linspace(a,b,n);
    ind = find(c <= x & x <= d);
    z = zeros(n,n); z(ind,ind) = 1;  // a step inside a square
    // create evaluation grid
    np = 220;
    xp = linspace(a,b, np);
    [XP, YP] = ndgrid(xp, xp);
    zp1 = interp2d(XP, YP, x, x, splin2d(x,x,z));
    zp2 = interp2d(XP, YP, x, x, splin2d(x,x,z,"monotone"));
    // plot
    xbasc()
    xset("colormap",jetcolormap(128))
    subplot(1,2,1)
       plot3d1(xp, xp, zp1, flag=[-2 6 4])
       xtitle("spline (not_a_knot)")
    subplot(1,2,2)
       plot3d1(xp, xp, zp2, flag=[-2 6 4])
       xtitle("subspline (monotone)")

### See Also

`cshep2d`, `linear_interpn`, `interp2d`,

### Author

B. Pincon